Graph the rational functions. Locate any asymptotes on the graph. f(x)=((x-1)(x^2-4))/((x-2)(x^2

step 1 Here we see the function, F of X is equal to, well, X minus 1 times the quantity X squared minus

Graph the rational functions. Locate any asymptotes on the...

Key Concepts

Rational Functions

A rational function is defined as the quotient of two polynomials. Its characteristics depend on the factors in the numerator and denominator, and the function's domain is determined by the values that do not make the denominator zero. Understanding rational functions involves examining their zeros, intercepts, and potential points of discontinuity.

Vertical Asymptotes

Vertical asymptotes occur at the values of the variable that make the denominator zero, provided that these values do not also cancel out with factors in the numerator. They indicate where the function tends toward positive or negative infinity, representing non-removable discontinuities in the graph.

Horizontal and Oblique Asymptotes

The end behavior of a rational function is often described by horizontal or oblique asymptotes. Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator, indicating the value that the function approaches as the variable tends to infinity. Oblique asymptotes occur when the degree of the numerator is exactly one higher than the denominator, representing a slanted line that the function approaches at the extremes.

Removable Discontinuities

Removable discontinuities, or holes, occur when a factor is common to both the numerator and the denominator. Although these factors may cancel out algebraically, they indicate points on the graph where the function is not defined, despite the overall behavior resembling a continuous function.

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